Hypothesis Testing – Distribution

Hypothesis TestingHypothesis Testing – Distribution

Hypothesis Testing – DistributionSTATISTICS – Lecture no. 12

Jirı Neubauer

Department of Econometrics FEM UO Brnooffice 69a, tel. 973 442029email:[email protected]

5. 1. 2010

Jirı Neubauer Hypothesis Testing – Distribution


Hypothesis Testing

A statistical hypothesis is a claim (or statement) about

a population parameters (µ, σ2, π, λ, . . . ),

a distribution (normal, Poisson, . . . ).



Hypothesis Testing

A null hypothesis H is a claim (or statement) abouta population parameter that is assumed to be true until it isdeclared false . . . for example

H : µ = µ0,

An alternative hypothesis A is a claim about a populationparameter that will be true if the null hypothesis is false

1. A : µ 6= µ0 → both-sided test,2. A : µ > µ0 → one-sided test,3. A : µ < µ0 → one-sided test.



Hypothesis Testing


H : µ = µ0,


1. A : µ 6= µ0 → both-sided test,

2. A : µ > µ0 → one-sided test,3. A : µ < µ0 → one-sided test.



Hypothesis Testing


H : µ = µ0,


1. A : µ 6= µ0 → both-sided test,2. A : µ > µ0 → one-sided test,

3. A : µ < µ0 → one-sided test.



Hypothesis Testing


H : µ = µ0,


1. A : µ 6= µ0 → both-sided test,2. A : µ > µ0 → one-sided test,3. A : µ < µ0 → one-sided test.



Hypothesis Testing

reality H is true H is false

decision about H prob. prob.

not reject correct decision 1− α type II error β

reject type I error α correct decision 1− β



Hypothesis Testing

If we reject the null hypothesis which is true, we call this typeI error. The probability of this error is α ⇒ a significancelevel. A number 1− α is probability that we do not reject thetrue hypothesis H.

If we accept the null hypothesis although is false, we call thistype II error. The probability of this error is β. A number1− β ⇒ a power of the test is the probability that we rejectthe null hypothesis H, if it is false.



Hypothesis Testing

To test the null hypothesis we use a function of a random sampleT = T (x1, x2, . . . , xn), so called test statistic, which has under thenull hypothesis H known distribution (usually t, u, χ2,F ).We divide the all possible values of the test statistic into

W1−α - nonrejection region of H – the set of valuesconnected with the hypothesis H,

Wα - rejection region of H – the set of values connectedwith the hypothesis A.



Steps to Perform a Test of Hypothesis

1. State the null and alternative hypothesis H and A.

2. Select a significance level α (usually 0.05 a 0.01).

3. Choose the test statistic.

4. Determine the rejection region Wα.

5. Calculate the value of the test statistic.




6. Make a decision:

If the value of the test statistic falls in the rejection region, wereject the null hypothesis H and say that we accept thealternative hypothesis A with the probability 1− α.If the value of the test statistic falls in the nonrejection region,we do not reject the null hypothesis H.




6. Make a decision:

If the value of the test statistic falls in the rejection region, wereject the null hypothesis H and say that we accept thealternative hypothesis A with the probability 1− α.

If the value of the test statistic falls in the nonrejection region,we do not reject the null hypothesis H.




6. Make a decision:

If the value of the test statistic falls in the rejection region, wereject the null hypothesis H and say that we accept thealternative hypothesis A with the probability 1− α.If the value of the test statistic falls in the nonrejection region,we do not reject the null hypothesis H.



Chi-Square Goodness of Fit TestTests of Skewness and KurtosisCompound Tests of Skewness and Kurtosis

Chi-Square Goodness of Fit Test

We divide values of a random sample x1, x2, . . . , xn into k disjunctclasses, where nj , j = 1, 2, . . . , k, is frequency of the class j and πj

is a probability that the random variable X has value from theclass j , calculated on condition that X has an assumed distribution.

The main idea of the test is to compare relative frequencies nj/nwith theoretical probabilities πj .





We state the null and alternative hypothesis:H : the random X has an assumed distribution → A : the randomX has not an assumed distribution.The test statistic is

χ2 =k∑

j=1

(nj − nπj)2

nπj,

which has under the null hypothesis H for large n (asymptotically)a Pearson χ2-distribution with ν = k − c − 1 degrees of freedom,where c is a number of estimated parameters of the assumeddistribution. A rejection region is

Wα ={χ2, χ2 ≥ χ2

1−α(ν)}

,

where χ21−α(ν) is a quantile of the Pearson χ2-distribution.





Recommendation:

nπj > 5, j = 1, 2, . . . , k.

If this condition is not satisfied, it is necessary to join the classes.




Tests of Skewness and Kurtosis

The normal distribution has α3 = 0 a α4 = 0. We can use theseproperties to test normality. We calculate a sample skewness andkurtosis (they are estimates of α3 and α4)

α3 = a3 =1

ns3n

n∑i=1

(xi − x)3, α3 = a4 =1

ns4n

n∑i=1

(xi − x)4 − 3.

We state hypothesis:H1 : α3 = 0 → A1 : α3 6= 0H2 : α4 = 0 → A2 : α4 6= 0





1. H1 : α3 = 0 → A1 : α3 6= 0Test statistic is

u3 =a3√D(a3)

, where D(a3) =6(n − 2)

(n + 1)(n + 3),

which has under the null hypothesis H1 asymptotically normaldistribution N(0, 1). A rejection region is

Wα ={

u3, |u3| ≥ u1−α2

},

where u1−α2

is a quantile of N(0, 1).





2. H2 : α4 = 0 → A2 : α4 6= 0Test statistic is

u4 =a4 + 6

n+1√D(a4)

, where D(a4) =24n(n − 2)(n − 3)

(n + 1)2(n + 3)(n + 5),

which has under the null hypothesis H2 asymptotically normaldistribution N(0, 1). A rejection region is

Wα ={

u4, |u4| ≥ u1−α2

},

where u1−α2

is a quantile of N(0, 1).




Compound Tests of Skewness and Kurtosis

We state hypothesis:H : a random variable X has a normal distribution → A : a randomvariable X has not a normal distribution.Test statistic is

C = u23 + u2

4 ,

which has under the null hypothesis H approximately χ2

distribution with two degrees of freedom.u3 and u4 are test statistics defined above. A rejection region is

Wα ={C ,C ≥ χ2

1−α(2)}

,

where χ21−α(2) is a quantile of the Pearson χ2-distribution.


Hypothesis Testing – Distribution

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